What is the definition of a heuristic function in the BayesChess paper?
By far the most common form of heuristic evaluation functions for Chess-playing (or, really, any game-playing) agents are simple linear functions. At least when we're talking about handcrafted features that's the case, of course all the hype with Deep Neural Networks in more recent years is different. So, when it's not specified in a paper like this exactly what their heuristic evaluation function looks like, you can relatively safely assume it's just a linear function.

With linear function, I mean that you have vectors of features $\boldsymbol{\phi}(s)$ for your states $s$, and a vector of weights $\boldsymbol{\theta}$, and the evaluation $f(s)$ of a state $s$ is simply given by the dot product (summing up all the multiplications of feature values with their corresponding weights):

$$f(s) = \boldsymbol{\phi}(s)^{\top} \boldsymbol{\theta} = \sum_i \phi_i(s) \times \theta_i(s),$$

where the subscript $i$ indicates taking the $i^{th}$ element of a vector.